Trigonometric Identities: Precalculus Math Worksheet – PDF Download

Precalculus Trig Identities Worksheet: problems 1–5

Problem 1: Reciprocal and quotient forms

Rewrite each expression using only sine and cosine.

1. \(\text{sec}\,\theta\)
2. \(\text{csc}\,\theta\)
3. \(\text{cot}\,\theta\)
4. \(\tan\theta\,\text{csc}\,\theta\)

Problem 2: Reciprocal values

Find the reciprocal function value.

1. \(\sin\theta=\frac{3}{5}\). Find \(\text{csc}\,\theta\).
2. \(\cos\theta=-\frac{5}{13}\). Find \(\text{sec}\,\theta\).
3. \(\tan\theta=\frac{7}{24}\). Find \(\text{cot}\,\theta\).
4. \(\tan\theta=-2\). Find \(\text{cot}\,\theta\).

Problem 3: Pythagorean forms

Use a Pythagorean identity to find each value.

1. \(\cos\theta=0.6\). Find \(\sin^{2}\theta\).
2. \(\sin\theta=-\frac{1}{2}\). Find \(\cos^{2}\theta\).
3. \(\tan\theta=3\). Find \(\text{sec}^{2}\theta\).
4. \(\text{cot}\,\theta=2\). Find \(\text{csc}^{2}\theta\).

Problem 4: Cosine from sine, Quadrant I

Angle \(\theta\) is in Quadrant I and \(\sin\theta=\frac{4}{5}\). Find \(\cos\theta\) and \(\tan\theta\).

Problem 5: Cosine from sine, Quadrant II

Angle \(\theta\) is in Quadrant II and \(\sin\theta=\frac{5}{13}\). Find \(\cos\theta\) and \(\tan\theta\).

Precalculus Trig Identities Worksheet: problems 6–10

Problem 6: Sine from cosine, Quadrant III

Angle \(\theta\) is in Quadrant III and \(\cos\theta=-\frac{8}{17}\). Find \(\sin\theta\) and \(\tan\theta\).

Problem 7: Sine and cosine from tangent

Angle \(\theta\) is in Quadrant IV and \(\tan\theta=-\frac{3}{4}\). Use \(1+\tan^{2}\theta=\text{sec}^{2}\theta\) to find \(\cos\theta\) and \(\sin\theta\).

Problem 8: All six functions

Angle \(\theta\) satisfies \(\sin\theta=-\frac{24}{25}\) and \(\cos\theta< 0\). Find the other five trigonometric functions of \(\theta\).

Problem 9: Simplify products

Simplify each expression to a single trigonometric function.

1. \(\sin\theta\,\text{cot}\,\theta\)
2. \(\cos\theta\tan\theta\)
3. \(\text{csc}\,\theta\tan\theta\)
4. \(1-\cos^{2}\theta\)

Problem 10: Simplify with Pythagorean identities

Simplify each expression.

1. \(\sin^{2}\theta+\cos^{2}\theta+\tan^{2}\theta\)
2. \((1-\sin\theta)(1+\sin\theta)\)
3. \(\frac{\text{sec}^{2}\theta-1}{\tan\theta}\)
4. \(\cos^{2}\theta(1+\tan^{2}\theta)\)

Precalculus Trig Identities Worksheet: problems 11–15

Problem 11: Identity or not?

The graph compares \(y=\sin x+\cos x\) with \(y=1\).

1. Explain why the graph shows that \(\sin x+\cos x=1\) is not an identity.
2. Show that it fails at \(x=\frac{\pi}{4}\).
3. Check that \(\sin^{2}x+\cos^{2}x=1\) holds at \(x=\frac{\pi}{3}\).

Problem 12: Simplify rational expressions

Simplify each expression.

1. \(\frac{\sin^{2}\theta}{1-\cos\theta}\)
2. \(\frac{\cos^{2}\theta}{1-\sin\theta}\)
3. \(\frac{1}{1-\sin\theta}+\frac{1}{1+\sin\theta}\)

Problem 13: Verify an identity: sum of fractions

Verify the identity \(\frac{\sin\theta}{1+\cos\theta}+\frac{1+\cos\theta}{\sin\theta}=2\,\text{csc}\,\theta\).

Problem 14: Verify an identity: tangent plus cotangent

Verify the identity \(\tan\theta+\text{cot}\,\theta=\text{sec}\,\theta\,\text{csc}\,\theta\).

Problem 15: Verify an identity: secant minus cosine

Verify the identity \(\frac{\text{sec}\,\theta-\cos\theta}{\tan\theta}=\sin\theta\).

Precalculus Trig Identities Worksheet: problems 16–20

Problem 16: Exact values with the sum formula

Find the exact value of each expression.

1. \(\sin 75^{\circ}\)
2. \(\cos 105^{\circ}\)
3. \(\tan 15^{\circ}\)

Problem 17: Exact values in radians

Write each angle as a sum or difference of special angles and find the exact value.

1. \(\sin\frac{\pi}{12}\)
2. \(\tan\frac{5\pi}{12}\)
3. \(\cos\frac{7\pi}{12}\)

Problem 18: Sum formulas from two triangles

Angle \(A\) is in Quadrant I with \(\sin A=\frac{3}{5}\). Angle \(B\) is in Quadrant II with \(\cos B=-\frac{5}{13}\). Find \(\sin(A+B)\) and \(\cos(A+B)\), then name the quadrant of \(A+B\).

Problem 19: Recognize the formula

Rewrite each expression as one trigonometric value and evaluate it.

1. \(\sin 50^{\circ}\cos 20^{\circ}-\cos 50^{\circ}\sin 20^{\circ}\)
2. \(\cos 80^{\circ}\cos 20^{\circ}+\sin 80^{\circ}\sin 20^{\circ}\)
3. \(\frac{\tan 40^{\circ}+\tan 5^{\circ}}{1-\tan 40^{\circ}\tan 5^{\circ}}\)
4. \(\sin\frac{2\pi}{9}\cos\frac{\pi}{9}+\cos\frac{2\pi}{9}\sin\frac{\pi}{9}\)

Problem 20: Prove shift identities

Use a sum formula to prove each identity.

1. \(\sin(\theta+\frac{\pi}{2})=\cos\theta\)
2. \(\cos(\theta+\pi)=-\cos\theta\)
3. \(\tan(\theta+\pi)=\tan\theta\)

Precalculus Trig Identities Worksheet: problems 21–25

Problem 21: Double-angle values

Angle \(\theta\) is in Quadrant IV and \(\sin\theta=-\frac{5}{13}\). Find \(\sin 2\theta\), \(\cos 2\theta\) and \(\tan 2\theta\).

Problem 22: Double-angle simplifications

Rewrite each expression as one value and evaluate it exactly.

1. \(2\sin 15^{\circ}\cos 15^{\circ}\)
2. \(\cos^{2}22.5^{\circ}-\sin^{2}22.5^{\circ}\)
3. \(1-2\sin^{2}\frac{\pi}{12}\)
4. \(\frac{2\tan 15^{\circ}}{1-\tan^{2}15^{\circ}}\)

Problem 23: Work backward from cos 2θ

Maya knows that \(\cos 2\theta=\frac{7}{25}\) and that \(\theta\) is in Quadrant I. Use \(\cos 2\theta=2\cos^{2}\theta-1\) and \(\cos 2\theta=1-2\sin^{2}\theta\) to find \(\cos\theta\), \(\sin\theta\) and \(\tan 2\theta\).

Problem 24: Soccer kick range

The range of a kicked ball is \(R=\frac{v^{2}\sin 2\theta}{g}\) with \(g=32\text{ ft/s}^{2}\). Jamal kicks a ball at \(v=40\text{ ft/s}\) with \(\sin\theta=0.6\) and \(\cos\theta=0.8\).

1. Find \(\sin 2\theta\) and the range.
2. Find the range for a \(45^{\circ}\) kick at the same speed.
3. How many feet farther does the \(45^{\circ}\) kick travel?

Problem 25: Ferris wheel doubling

A Ferris wheel has radius 50 ft and its center is 60 ft above the ground. Ethan sits at angle \(\theta\) with his seat 30 ft above the center and to the left of it. The wheel turns until his angle is \(2\theta\).

1. Find \(\cos\theta\) and \(\sin\theta\).
2. Use double-angle formulas to find his new coordinates relative to the center.
3. How high above the ground is he at the new position?

Precalculus Trig Identities Worksheet: problems 26–28

Problem 26: Rotating a point

A graphics app rotates a point \(P(x,y)=(5\cosA,5\sinA)\) counterclockwise about the origin by \(30^{\circ}\). Mei starts with \(P(4,3)\).

1. Find \(\cos A\) and \(\sin A\).
2. Use the sum formulas to find the exact coordinates of the rotated point \(P^{\prime}\).
3. Show that \(P^{\prime}\) is still 5 units from the origin (use decimals).

Problem 27: Camera tilt

Diego tilts a drone camera upward in two steps. The first tilt is angle \(A\) with \(\tan A=\frac{1}{2}\) and the second is angle \(B\) with \(\tan B=\frac{1}{3}\). Both are acute.

1. Find \(\tan(A+B)\).
2. Find the total tilt \(A+B\) in degrees.
3. Find \(A\) and \(B\) separately to the nearest hundredth of a degree and check the sum.

Problem 28: Polarizing filter

Light passing through a polarizing filter has intensity \(I=80\cos^{2}\theta\) lumens. Aisha rewrites it with \(\cos^{2}\theta=\frac{1+\cos 2\theta}{2}\).

1. Show that this rewrite follows from \(\cos 2\theta=2\cos^{2}\theta-1\).
2. Find \(I\) for \(\theta=30^{\circ}\).
3. Find the exact intensity for \(\theta=15^{\circ}\) and round to the nearest hundredth.
4. Find the exact intensity for \(\theta=22.5^{\circ}\) and round to the nearest hundredth.

How to use this precalculus trig identities worksheet

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